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IN  MEMORIAM 
I     FLORfAN  CAJORI 


IV 


UMREB 


^  mathematical  ^ 
\    Series     M 


Vi         TWO 


^ 


^M  Tt  ^ 


lementary  Arithmetic 


or     THE 


Octimal  Notation 


By  GEO.  H.  COOPER 


PublisHed    by 
TMi:    \\'HITAIii:R     Ca    ray     CO. 

(iNCOnPOnATEo) 

£L<ivicational      PviblisHer? 
SAN     FRANCISCO 


X^  4-.  J^.A^ 


c^ 


ELEMENTARY  ARITHMETIC 


OP   THE 


OGTIMAL    NOTATION 


BY 
GEORGE    H.l  COOPER 


SAN   FRANCISCO 

THE    WHITAKER   AND   RAY   COMPANY 
(INCORPORATED) 

1902 


CAJORI 


Copyright,  1902 
By  George  H.  Cooper 


PUBLISHER'S  NOTE. 

This  book  is  paged  according  to  the  Octimal  System  of 

Numbers. 

ERRATA. 

Page  22.     Lines  3  and  4  should  read  :  "4  from  13  leaves  7.    Write 

down  7,"  etc. 
Page  51.     Line  3  should  read  :  "  Int.  3d  yr." 
Page  66.     Last  line  Miscellaneous.    Answer  should  read:  "  17.726 

Centimeters." 


PREFACE. 

History  shows  that  many  unsuccessful  attempts  have 
been  made  to  discover  a  system  of  notation  which  would 
be  simple  and  effective,  and  we  wonder  that  a  satisfac- 
tory basis  has  not  been  found.  It  is  the  more  surpris- 
ing, when  we  consider  that  every  part  of  the  universe 
suggests  multiplication  or  division  by  two  as  a  basis  from 
which  to  start  the  building  of  a  system  of  notation.  This 
principle  is  exposed  in  the  attempt  to  provide  a  Decimal 
currency,  for  the  result  is  an  Octimal  coinage,  viz.,  the 
Dollar,  half,  and  quarter.  And  so,  also,  the  attempt  to 
decimalize  the  circle,  for  the  quadrant  must  be  found  by 
the  same  process.  In  fact,  there  is  nothing  in  the  uni- 
verse to  suggest  the  use  of  ten  as  a  radix.  So  many  ob- 
stacles stand  in  the  way  of  the  successful  application  of 
the  Decimal  Notation,  that  it  is  astonishing  it  has  not 
not  been  abandoned  long  ago. 

The  Octimal  System  will  be  the  means  of  consider- 
ably reducing  the  work  in  schools,  by  rendering  much  of 
the  work  unnecessary,  which,  owing  to  the  defects  of 
the  Decimal  System,  is  now  required,  and,  in  addition, 
will  make  arithmetical  work  a  pleasure  instead  of  a 
drudgery. 

Its  introduction  need  cause  no  confusion,  for  the  pres- 
ent units  in  use  will  be  retained;  viz.,  the  Inch,  the 
Pound,  and  the  Dollar,  while  the  methods  of  manipula- 
tion are  the  same  as  those  already  practiced. 

Any  attempt  to  perpetuate  the  use  of  the  Decimal  Sys- 

3 


;n;oo4nno 


4  Preface. 

tern  is  nothing  short  of  a  crime  against  humanity,  since 
it  fails  in  every  department. 

The  principle  of  "metric"  arrangement  has  been 
adopted  in  this  work,  and  followed  out  to  its  logical  con- 
clusion, and  the  desired  result  has  been  attained,  —  a  per- 
fect system  of  notation,  and  arrangement  of  weights  and 
measures. 

The  Arithmetic  is  therefore  offered  in  the  confidence 
that  it  will  prove  an  educational  factor  of  great  value,  and 
forever  place  the  science  of  numbers  on  a  sound  basis. 

The  Author. 


CONTENTS. 

SECTION    1. 

Defixitioxs,  Xotatiox,  axd  Xumeratiox 7-13 

NoTATiox  Table 7 

NuMERATiox'  Table 11 

SECTION   2. 

Additiox ''. 15 

Additiox  Table 15 

Subtraction 20 

Multiplication 23 

Multiplication  Table 23 

Division 27 

SECTION    3. 

Definitions 34 

Reduction  35 

Weights,  Measures,  Money,  Time.     Tables 35 

Compound  Numbers  40 

Compound  Additiox 40 

CoMPouxD  Subtraction' 41 

Compound  Multiplication 41 

Compound  Division 42 

Repetends  43 

SECTION   4. 

Ratio  and  Proportion 44 

Rule  of  Three 45 

Practice.     Simple 46 

Practice.     Compound 47 

Interest.     Simple 47 

Interest.     Compound 50 

5 


6  Contents. 

SECTION    5. 

Square  Root 52 

Rectangle 54 

Cube  Root 55 

Rectangle  Parallelopiped 60 

Derivation  of  Radix 62 

Elevation  of  Denominations 62 

Elevation  of  Cubes  and  their  Roots 63 

Calculation  by  Construction 63 

SECTION   6. 

Longitude  and  Time 64 

To  Determine  an  Angle 64 

Area  of  Circle  in  Terms  of  Square  Units  65 

SECTION   7. 

Miscellaneous 66 

Transformation  of  Numbers 66 

Decimal  Fractions  —  To  Transform  into  Inferior  Units  ...  67 
Decimal  Numbers  and  Octimal  Inferior  Units.    70 


ELEiAlENTARY  ARITHMETIC 


OF   THE 


OCTIMAL    NOTATION. 

SECTION  1. 

ARITHMETIC. 

Arithmetic  teaches  us  the  use  of  Numbers. 

A  Unit,  or  One,  is  any  single  object  or  thing;  as,  an 
orange,  a  tree. 

A  Whole  Number,  or  an  Integer,  is  a  Unit,  or  One,  or 
a  collection  of  ones.  If  a  boy,  for  instance,  have  one 
orange,  and  then  another  is  given  to  him,  he  will  have  two 
oranges;  if  another  be  given  to  him,  he  will  have  three 
oranges;  if  another,  four  oranges,  and  so  on.  One,  two, 
three,  four,  etc.,  are  called  Whole  Numbers,  or  Integers. 

Notation  is  the  art  of  writing  any  number  in  figures. 

The  Octimal  Notation  is  the  method  of  expressing  num- 
bers by  means  of  the  following  radix  of  figures:  — 

12         3        4        5      6        7  0 

called   one,  two,  three,  four,  five,  six,  seven,  eights,  or  cipher. 

representing  (if  we  express  a  unit  by  a  dot;  thus,  .), 


or  one   or  two   or  three  or  four   or  five  or  six   or  seven   or  eights 
unit       units       units       units     units     units       units         units 

The  first  seven  figures  have  a  fixed  value.  The  value 
of  the  last  depends  upon  whether  it  is  prefixed  by  another 
figure  or  not,  and  also  the  value  of  the  prefixed  figure. 
When  standing  alone,  without  an  index,  it  has  no  value; 
but  when  prefixed  by  the  Index,  one,  its  value  is  eight;  if 

7 


10      Elementary  Arithvietic  of  the  Octimal  Notation. 

by  two,  two  eights  or  two-et,  etc.,  etc.  Any  of  the  figures 
from  one  to  seven,  when  prefixed  to  the  cipher  (thus,  10), 
indicates  that  the  radix  has  been  named  over  as  many 
times  as  the  prefixed  figure  has  value  in  units.  Thus  if  the 
radix  has  been  counted  through  once,  the  figure  1  is  to  be 
prefixed;  and  if  twice,  the  figure  2;  and  if  three  times, 
the  figure  3,  and  so  on  until  all  the  figures  have  been  pre- 
fixed. 

The  order  of  progression  in  the  Octimal  System  of  No- 
tation is  in  powers  of  eight;  that  is  to  say,  that  when 
eight  has  been  counted  once,  that  fact  is  to  be  recorded  by 
prefixing  the  figure  1  to  the  cipher  (thus,  10),  so  that 
while  the  figures  of  fixed  value  stand  alone,  they  only 
stand  for  the  number  of  units  they  represent;  but  the 
addition  of  the  cipher  denotes  that  the  number  to  which 
it  is  affixed  has  been  counted  as  many  eight  times. 
Thus  70  denotes  that  7  has  been  counted  eight  times,  or 
8  seven  times,  and  the  figure  prefixed  to  the  cipher  to 
record  that  fact. 

When  the  radix  has  been  counted  through  eight  times, 
the  figures  which  stand  for  one  eight  are  to  be  prefixed  to 
a  new  cipher;  thus,  100,  one  eight  eight,  the  name  of  this 
combination  being  abbreviated  to  etred.  All  numbers 
greater  than  eight,  but  less  than  etred,  will  be  repre- 
sented by  two  figures;  thus  11  signifies  that  eight  has 
been  counted  once,  because  the  figure  1  appears  as  the 
second  figure  to  the  left,  the  place  to  be  occupied  by  the 
figures  which  represent  the  number  of  eights  which  have 
been  counted,  and  the  figure  1  to  the  right  denotes  that 
one  more  than  eight  has  been  counted;  and  as  the  figure 
1  must  occupy  the  units'  place  in  the  number,  the  cipher 
must  be  replaced  by  that  figure. 


Arithmetic. 


11 


We  therefore  understand  that  the  first  place  to  the  left 
of  a  point  belongs  to  the  ones  of  units  counted  up  to 
seven  and  the  cipher  when  the  number  is  a  multiple  of 
eight,  and  the  second  place  to  the  left  belongs  to  the 
eights  of  units  up  to  the  seventh  eight,  and  a  cipher 
when  the  number  is  a  multiple  of  eight,  and  so  on. 


Example. 


Thus  one 


is  written 


a 


eight 

etred 

etand 

eight  etand 

etred  etand 

million 

eight  million  " 

etred  million  " 

billion  " 


u 


u 


1. 

10. 

100. 

1,000. 

10,000. 

100,000. 

1,000,000. 

10,000,000. 

100,000,000. 

1,000,000,000. 


The  reversal  of  positions  of  figures  alters  their  value. 


Thus  one  eighth 
one  etredth 
one  etandth 
one  eight  etandth 
one  etred  etandth 
one  millionth 
one  eight  millionth  " 
one  etred  millionth  " 
one  billionth  " 


is  written  .1,     or  point  1. 
.01,    "       "      01. 
.001, "       '^      001. 
.0001,    and  so  on. 
.00001. 
.000001. 
.0000001. 
.00000001. 
.000000001. 


ii 


u 


u 


u 


u 


(( 


From  the  above  table  we  see  that  dividing  any  number 


12      Elementary  Arithmetic  of  the  Ortimal  Notation. 

into  periods  of  three  figures  each,  beginning  at  the  right 
hand,  the  names  of  those  periods  will  be  — 

First     period,  Units. 
Second       "        Etands. 
Third        "        MilHons. 
Fourth      "       Billions. 
Fifth  "       Trillions. 

The  names  of  the  places  in  each  of  these  periods  are  the 
same. 

Namely,  First      place.  Units. 
Second      "       Eights. 
Third         "       Etreds. 

The  following  plan  is  recommended  to  enable  the 
pupil  to  write  in  figures  any  number  dictated  by  the 
teacher:  — 

Let  the  pupil  write  on  his  slate  a  number  of  ciphers; 
thus,  000,000,000,000,  marking  them  off  into  periods  of 
three  places  each  from  the  right; 

Put  U  over  the  first     period  for  Units; 
E      '•       "    second      "        "    Etands; 
M     ''      "    third         "        "    Millions; 
B      "      "    fourth      "        "    Billions; 

B     M    E     U 

and  so  on;  thus,  000,000,000,000.  Then  when  a  number 
is  dictated  to  a  pupil,  all  he  has  to  do  is  to  put  each  fig- 
ure in  its  proper  place  and  fill  up  vacancies  with  O's. 


Arithmetic,  13 


Exercises. 


Write  the  following  numbers  in  figures:  — 

(1)  One,  four,  six,  three,  seven,  eight. 

(2)  Et-seven,  two-et,  four-et  one,  six-et  six,  two-et  two. 

(3)  Seven-et  three,  etred  and  et-one,  three  etred  and 
two-et  two. 

(4)  Etand  six  etred  and  four-et  three,  three  etand  and 
five. 

(5)  Seven-et  six  etand  three  etred  and  five-et  six. 

(6)  Two-et  three  million  seven  etred  and  six-et  etand 
and  four. 

(7)  Four  etred  million  three  etred  etand  five  etred  and 
two-et. 

Thus  two  etand  and  three  will  be  written  thus  in 
figures:  2,003; 

Six-et  etand,  three  etred  and  two:  60,302; 

Four  etred  and  three-et  etand,  six  etred  and  two-et  one: 
430,621; 

Six  etred  and  one  millions,  four  etred  and  seven:  601,- 
000,407. 

Numeration  is  the  art  of  waiting  in  words  the  meaning 
of  any  number  which  is  already  given  in  figures;  thus, 

35  means  three  eights  and  five  units,  or  three-et  five; 

506  means  five  etreds  no  eights  and  six  units,  or  five 
etred  and  six; 

0604  means  no  etands  six  etreds  no  eights,  or  six  etred 
and  four; 

3,251,630  means  three  millions,  two  etred  and  five-et 
one  etands,  six  etred  and  three-et. 


14      Elementary  Arithmetic  of  the  Octimal  Notation. 

Exercises. 

Write  in  words  the  meaning  of — 

(1)  2,  4,6,  1,  7,  12,  16,  40,  23,  75,  67,  32,  44. 

(2)  63,  21,  17,  35,52,  47,  77,  34,  16,  75. 

(3)  102,  165,  143,  371,  263,  542,  764,  463. 

(4)  1423,  3764,  4321,  5635,  7352,  6473. 

(5)  53742,  64371,  42536,  57324,  65437. 

(6)  140526,  330000,  4773254,  4613025. 


Addition. 


15 


SECTION  2. 

ADDITION. 

Addition  is  the  method  of  finding  a  number  which  is 
equal  to  two  or  more  numbers  of  the  same  kind  taken 
together. 

The  numbers  to  be  added  are  called  Addends. 

The  Sum,  or  Amount,  is  the  number  so  found. 


ADDITION 

TABLE. 

2  and 

3  and 

1  make 

3,  Three. 

1  make    4,  Four. 

2      " 

4,  Four. 

2      " 

5,  Five. 

3      " 

5,  Five. 

3      " 

6,  Six. 

4      " 

6,  Six. 

4      " 

7,  Seven. 

5      " 

7,  Seven. 

5      " 

10,  Eight. 

6      " 

10,  Eight. 

6      " 

11,  Et-one. 

7      " 

11,  Et-one. 

7      " 

12,  Et-two. 

10      " 

12,  Et-two. 

10      ' 

13,  Et-three. 

4  and 

5  and 

1  make 

5,  Five. 

1  mj 

ike    6,  Six. 

2      " 

6,  Six. 

2 

"        7,  Seven. 

3      " 

7,  Seven. 

3 

"      10,  Eight. 

4      " 

10,  Eight. 

4 

"      11,  Et-one. 

5      " 

11,  Et-one. 

0 

"      12,  Et-two. 

6      " 

12,  Et-two. 

6 

"      13,  Et-three. 

7      " 

13,  Et-three. 

"      14,  Et-four. 

10      " 

14,  Et-four. 

10 

"      15,  Et-five. 

16      Elementary  AritJivietic  of  the  Octimal  Notation. 


6  and 

1  make 

2 

3 

4 

5 

6 


u 


li 


u 


i 
10 


u 


it 


7,  Seven. 

10,  Eight. 

11,  Et-one. 

12,  Et-two. 

13,  Et-three. 

14,  Et-four. 

15,  Et-five. 

16,  Et-six. 


7  and 

1  make  10,  Eight. 

2  "  11,  Et-one. 

3  "  12,  Et-two. 

4  "  13,  Et-three. 

5  ''  14,  Et-four. 

6  "  15,  Et-five. 

7  "  16,  Et-six. 
10      "  17,  Et-seven. 


The  sign  +  called  plus,  placed  between  two  numbers 
means  that  the  two  numbers  are  to  be  added  together; 
thus  2  +  2  make  4,  and  2  +  2  +  1  make  5. 

The  sign  =  placed  between  two  sets  of  numbers  means 
that  they  are  equal  to  one  another;  thus,  2+2  =  4,  and 

2  +  2  +  1=4  +  1,  and  2  +  2  +  1=5,  and  4  +  1=5.     The  sum 
of  2  +  2  +  1  and  4  +  1  are  equal. 

The  sign  .".  means  therefore. 

Example  1. — ^Find  the  sum  of  3,  5,  and  2.  We  add 
thus:  3  and  5  make  10,  10  and  2  make  12;  .'.  the  sum 
of  3  +  5  +  2  =  12. 

Example  2.  —  Add  together  5,  7,  2,  6,  and  3.  5  and  7 
make  14,  14  and  2  make  16,  16  and  6  make  24,  24  and 

3  make  27;  .-.  5  +  7  +  2  +  6  +  3  =  27; 


or  thus:  — 


Addition.  17 

Add-     (1)         (2)         (3)         (4)         (5) 

4  3  4  7 


o 
O 


5  6  7  0  3 
2  2  112 
7  5  0            2            5 

6  3  6  5  1 

27  24  21  14  22 

RULE    FOR    ADDITION. 

Rule.  —  Write  down  the  given  numbers  under  each  other,  so 
that  units  may  come  under  units,  eights  under  eights,  etreds 
under  etreds,  and  so  on  ;  then  draw  a  line  under  the  lowest  num- 
ber. 

Find  the  sum  of  the  column  of  units  ;  if  it  be  less  than  eight, 
write  it  down  under  the  column  of  units,  below  the  line  just 
drawn;  but  if  it  be  greater  than  eight,  or  a  multiple  of  eight, 
then  write  down  the  surplus  of  units  under  the  units'  column, 
and  carry  to  the  column  of  eights  the  remaining  figure  or  figures, 
which  of  course  will  be  eights. 

Add  the  column  of  eights  and  the  figure  or  figures  you  carry 
as  you  have  added  the  column  of  units  and  treat  its  sum  in  the 
same  way  you  have  treated  the  column  of  units. 

Treat  each  succeeding  column  in  the  same  way. 

Write  down  the  full  sum  of  the  last  column  on  the  left  hand. 

The  entire  sum  thus  obtained  will  be  the  sum  or  amount  of 
the  given  numbers. 

Example  1.  —  Add  toi^ether  46,''53,  and  164. 

~  7  7 

By  the  Rule.  46         Method  of  Adding.  — 4  and  3  are  7,  7 

r  q 

and  6  are  15.    Write  down  5  and  carry  1, 
because  the  sum  of  the  units'  column 
305     is  one  eight  units  and  5  units  over. 


20      Elementary  Arithmetic  of  the  Octimal  Notation. 

Then  1  and  6  are  7,  7  and  5  are  14,  14  and  4  are  20. 
Write  down  0  under  the  eights'  column  and  carry  2. 
Then  2  and  1  are  3.  Write  down  3  under  the  etreds' 
cohimn,  which  completes  the  calculation. 


Examples. 

-  (1) 

(2) 

(3) 

(4) 

(5) 

(6) 

11 

15 

/ 

21 

35 

56 

13 

6 

16 

32 

47 

44 

21 

23 

14 

16 

23 

73 

45 

46 

41 

71 

127 

215 

(7) 

(10) 

(11) 

(12)  1 

(13) 

(14) 

52 

76 

134 

73 

321 

530 

73 

121 

62 

252 

763 

627 

104 

32 

251 

4 

222 

134 
501  ; 

420 

46 

251 

L724 

1425 

(15) 

(16) 

(17) 

(20) 

7350 

1276 

6325 

0 

37654 

14263 

43502 

52403 

263421 

54301 

73025 

73526 

4 

37562 

100134         140025         154456       1463057 


SUBTRACTION. 

Subtraction  is  the  method  of  finding  what  number  re- 
mains when  a  smaller  number  is  taken  from  a  greater 
number. 


Subtraction.  21 

The  number  so  found  is  called  the  remainder,  or  differ- 
ence. 

The  sign  — ,  called  Minus,  placed  between  two  numbers, 
means  that  the  second  number  is  to  be  subtracted  from 
the  first  number;  thus  7  —  3  means  that  3  is  to  be  sub- 
tracted from  7;  '.*.  7—3=4. 

RULE   FOR   SUBTRACTION. 

Rule.  — Write  down  the  less  number  under  the  greater  number, 
so  that  units  may  come  under  units,  eights  under  eights,  etreds 
under  etreds,  and  so  on;  then  draw  a  straight  line  under  the 
lower  number. 

Take,  if  you  can,  the  number  of  units  in  each  figure  of  the 
lower  number  from  the  number  of  units  in  each  figure  of  the 
upper  number  which  stands  directly  over  it,  and  place  the  re- 
mainder under  the  line  just  drawn,  units  under  units,  eights 
under  eights,  etreds  under  etreds,  and  so  on. 

But  if  the  units  in  any  figure  in  the  lower  number  be  greater 
than  the  number  of  units  in  the  figure  just  above  it,  then  add 
eight  to  the  upper  figure,  and  then  subtract  the  number  of  units 
in  the  lower  figure  from  the  number  of  units  in  the  upper  figure 
thus  increased,  and  write  down  the  remainder  as  before. 

Add  one  to  the  next  number  in  the  lower  number,  and  then 
take  this  figure  thus  increased  from  the  figure  just  above  it,  by 
one  of  the  methods  already  explained. 

Go  on  thus  with  all  the  figures. 

The  whole  difference,  or  remainder,  so  written  down  will  be 
the  difference,  or  remainder,  of  the  given  numbers. 

Example  1.  —  Subtract  3426  from  5345. 

By  the  Rule.  5345  Method.  — 6  from  5  I  cannot  take; 

3426     •'•  I  borrow  10.     Now  10  +  5  =  15.     I 

T^. ».  .^.  „     take  6  from  15,  which  leaves  7.    Write 

Dmerence=l/1  /       ,  ,,        ^  ,  ^       -..-r 

down    the     /     and    carry    1.     ^ext, 


22       Elementary  Arithinetic  of  the  Octimal  Notation. 

1  +  2  =  3.  Take  3  from  4,  which  leaves  1.  Write  down 
1  in  the  eights'  place.  Now  4  from  3  I  cannot  take.  I 
borrow  10.  Now  10  +  3=13.  4  from  13  leaves  5.  Write 
down  5  in  the  etreds'  place  and  carry  1.  Next,  1+3=4. 
Take  4  from  5,  which  leaves  1.  Write  down  1  in  the 
etands'  place. 

XoTE.  —  The  truth  of  all  sums  in  subtraction  may  be  proved  by 
adding  the  less  number  and  the  remainder  together.  If  the  sum 
has  been  worked  correctly,  the  two  numbers  will  be  equal. 

Thus  proof  of  Example  1 :  Less  number  +  remainder  =:  3426  + 
1717  -=5345,  the  greater  number. 


♦ 

Examples. 

(1) 

(2) 

(3) 

(4) 

(5) 

(<3) 

(7) 

From 

20 

24 

17 

73 

21 

75 

76 

Subtract  16 

13 

3 

14 

16 

37 

63 

02 

11 

14 

57 

03 

36 

13 

(10) 

(11) 

(12) 

(13) 

(14) 

(15) 

(16) 

265 

327 

630 

746 

642 

525 

740 

124 

146 

245 

362 

137 

431 

713 

141         161         363         364         503         074         025 


(17) 

(20) 

(21) 

(22) 

(23) 

(24) 

2653 

5341 

3421 

4763 

371246 

576356 

1241 

1432 

2675 

3652 

246517 

537642 

1412  3707  0524  1111  122527  036514 


Multiplication.  23 

MULTIPLICATION. 

Multiplication  is  a  short  method  of  repeated  addition; 
thus  when  2  is  multiplied  by  3,  the  number  obtained  is 
the  sum  of  2  repeated  3  times,  which  sum  =  2 -f  2  4-2  =  6. 

The  number  which  is  to  be  repeated  or  added  to  itself 
is  called  the  Multiplicand;  thus  in  the  above  example,  2 
is  the  multiplicand. 

The  number  which  shows  how  often  the  multiplicand 
is  to  be  repeated  is  called  the  Multiplier;  thus  in  the 
above  example,  3  is  the  multiplier. 

The  number  found  by  multiplication  —  for  instance,  6 
in  the  above  example  —  is  called  the  Product. 

The  multiplier  and  multiplicand  are  sometimes  called 
Factors,  because  they  are  factors,  or  makers,  of  the 
product. 

The  sign  X,  called  Into,  or  Multiplied  by,  placed  be- 
tween two  numbers,  means  that  the  two  numbers  are  to 
multiplied  together. 

The  following  Multiplication  Table  should  be  com- 
mitted to  memory:  — 


Twice 

3  times 

1  are  2,  Two. 

1  are  3,  Three. 

2    '^     4,  Four. 

2    "     6,  Six. 

8    '^     6,  Six. 

3    "   11,  Et-one. 

4   "  10,  Eight. 

4    "  14,  Et-four. 

5   "  12,  Et-two. 

5   "  17,  Et-seven. 

6   "  14,  Et-four. 

6   "  22,  Two-et  two, 

7   "  16,  Et-six. 

7    "  25,  Two-et  five. 

10   "  20,  Two-et. 

10   "  30,  Three-et. 

24       Elementary  Arithmetic  of  the  Octimal  Notation, 


4  times 

5  times 

1  are  4,  Four. 

1  are  5,  Five. 

2    "  10,  Eight. 

2    "  12,  Et-two. 

3   "  14,  Et-four. 

3   '•  17,  Et-seven. 

4   ''  20,  Two-et. 

4    "  24,  Two-et  four. 

5    "  24,  Two-et  four. 

5    ''  31,  Three-et  one. 

6   "  30,  Three-et, 

6    "  36,  Three-et  six. 

7    "  34,  Three-et  four. 

7    "43,  Four-et  three 

10   "  40,  Four-et. 

10   "  50,  Five-et. 

6  times 

7  times 

1  are  6,  Six. 

1  are  7,  Seven. 

2   "  14,  Et-four. 

2    "  16,  Et-six. 

3   "22,  Two-et  two. 

3    "  25,  Two-et  five. 

4   "  30,  Three-et. 

4   "  34,  Three-et  four, 

5   "  36,  Three-et  six. 

5    "  43,  Four-et  three, 

6    "  44,  Four-et  four. 

6    "  52,  Five-et  two. 

7   "  52,  Five-et  two. 

7    "  61,  Six-et  one. 

10   "  60,  Six-et. 

10   "  70,  Seven-et. 

10  times 

10  times 

10  are  100,  Etred. 

100  are  1,000,  Etand. 

1,000  times 

1,000  are  1,000,000,  Million. 

RULE  FOR  MULTIPLICATION  WHEN  THE  MULTIPLIER 
IS   NOT   GREATER   THAN   SEVEN. 

Rule.  —  Place  the  multiplier  in  the  units'  place  under  the 
multiplicand.  Draw  a  line  under  the  multiplier.  Multiply  each 
figure  of  the  multiplicand,  beginning  with  the  units,  by  the  fig- 


Multiplication.  25 

lire  of  the  multiplier  (by  means  of  the  ^Multiplication  Table). 
Write  down  and  carry  as  in  Addition. 

Example  1.  —  Multiply  325  by  2. 

By  the  Rule.  325  Twice  5  units  makes  12  units.    Write 

2     2  in  the  units'  place  in  the  product, 

g-9     and  carry  1.     Next,  twice  2  makes  4. 

Add  to  the  4  the  1  wliich  was  carried, 

which  makes  5.    Write  down  5  in  the  eights'  place.    Next, 

twice  3  makes  6.     Write  the  6  down  in  the  etreds'  place. 


Examples. 

(1) 

{'2) 

(3) 

(4) 

Multiply    231 

357 

426 

734 

Bv                   2 

3 

o 

6 

462     1315  2556  5450 

(5)      (6)  (7)  (10) 

7342     6321  52264  563421 

5.7  3  4 


45152    54667    177034    2716104 

RULE    FOR    MULTIPLICATION    AVHEX    MULTIPLIER    IS 
A   NUMBER   LARGER  THAN   SEVEN. 

Rule.  —  Place  the  multiplier  under  the  multiplicand,  units 
under  units,  eights  under  eights,  etreds  under  etreds,  and  so 
on ;  then  draw  a  line  under  the  multiplier.  Multiply  each  fig- 
ure in  the  multiplicand,  beginning  with  the  units,  by  the  figure 
in  the  units'  place  of  the  multiplier  (by  means  of  the  Multiplica- 
tion Table) ;  write  down  and  carry  as  in  Addition. 

Then  multiply  each  figure  of  the  multiplicand,  beginning  with 
the  units,  by  the  figure  in  the  eights'  place  of  the  multiplier, 


26       Elementary  Arithmetic  of  the  OrtiriKil  Notation. 

placing  the  first  figure  h^o  obtained  under  the  eights  of  the  line 
above,  the  next  figure  under  the  etreds,  and  so  on.  Proceed  in 
the  same  way  with  each  succeeding  figure  of  the  multiplier. 

Then  add  up  all  the  results  thus  obtained,  by  the  rule  of 
Addition. 

Example  2.  — Multiply  4362  by  342. 

4362 
342  Since  342  =  300  +  40  +  2,  we  really  multiply 

.        first  by  2,  next  by  40,  and  last  by  300,  and  if 

^  the  sums  are  placed   under  each  other  and 

added   together,   the   product  will   appear  as 

shown. 

1762644 

To  multiply  by  eight,  all  that  is  necessary  is  to  affix 
a  cipher  to  the  right  of  the  number,  and  by  etred,  two 
ciphers,  and  so  on. 

When  any  other  Multiplier  terminates  with  one  or 
more  ciphers,  multiply  by  the  remaining  figures,  and  to 
the  product  add  the  same  number  of  ciphers;  thus:  — 

Example  3,  — Multiply  42650  by  2300. 

42650 
2300 


15037 
10552 

122557000 


Multiplication  by  6  may  be  accomplished  with  the 
same  result  by  its  factors  2  and  3  in  succession,  and  so 
with  the  factors  of  any  other  number. 


Division 


27 


Numbers  which  cannot  be  broken  up  into  factors  are 
called  Prime  Numbers,  —  3,  5,  7,  13,  etc. 


Examples. 

(1) 

{'^) 

(3) 

(4) 

Multiply  3642 

5340 

4263 

57312 

By      231 

132 

246 

6421 

3642 

12700 

32062 

57312 

13346 

20240 

21314 

136624 

7504 

5340 

10546 

275450 

1107722 

751300 

1322022 

434274 

465506552 

DIVISION 

• 

Division  is  a  short  method  of  repeated  subtraction;  or, 
it  is  the  method  of  finding  how  often  one  number,  called 
the  Divisor,  is -contained  in  another  number,  called  the 
Dividend.  The  number  which  shows  this  is  called  the 
Quotient. 

Thus  the  dividend  11  divided  by  the  divisor  3  gives 
the  quotient  3;  and  for  this  reason  3  +  3  +  3  =  11,  and 
therefore  if  we  subtract  3  from  11,  and  then  a  second  3 
from  the  remainder  6,  and  then  a  third  3  from  the  re- 
mainder 3,  nothing  remains. 

If,  however,  some  number  be  left  after  the  divisor  has 
been  taken  as  often  as  possible  from  the  dividend,  that 
number  is  called  the  Remainder;  thus  10  divided  by  3 
gives  a  quotient  2,  and  a  remainder  2;  for,  after  subtract- 


30      Elementary  Arithmetic  of  the  Octimal  Notation. 

ing  3  from  10  once,  there  is  a  reroainder  5;  after  subtract- 
ing 3  a  second  time  from  the  remainder  5,  there  is  a 
remainder  2. 

The  sign  h-,  called  By,  or  Divided  by,  placed  between 
two  numbers,  signifies  that  the  first  figure  is  to  be  divided 
by  the  second. 

Division  is  the  opposite  of  multiplication.  By  the  Mul- 
tiplication Table,  3X4-14,  and  14-^4=3,  or  14^3=4. 

RULE  FOR  DIVISION  WHEN  THE  DIVISOR  IS  A    NUM- 
BER  NOT   LARGER   THAN   SEVEN. 

Rule.  —  Place  the  divisor  and  dividend  thus  :  — 

divisor  )  dividend 
quotient 

Take  off  from  the  left  hand  of  the  dividend  the  least  number 
of  figures  which  make  a  number  not  less  than  the  divisor.  Find 
by  the  Multiplication  Table  how  often  the  divisor  is  contained  in 
this  number ;  write  the  quotient  under  the  units'  figure  of  this 
number;  if  there  is  a  remainder,  affix  to  it,  in  the  units'  place, 
the  next  figure  in  the  dividend,  and  proceed  as  before.  AVhen- 
ever  there  is  no  remainder,  and  the  next  figure  does  not  contain 
the  divisor,  place  a  cipher  in  the  next  place  to  be  filled  in  the 
quotient.  If  there  be  a  remainder  at  the  end  of  the  operation, 
write  it  distinct  from  the  quotient,  and  write  the  divisor  under  it 
for  a  record. 

Example  1. —  Divide  1652  b}^  4. 

By  the  Rule.  4)16d2  Method.  —  4  into   1    I   cannot; 

oroi     therefore  affix  the  next  figure  to 

the  1.     Then  4  into  16  will  go  3 

times,  leaving  a  remainder  2;   wa-ite  down  the  3  under 


Division.  31 

the  units'  place  of  the  number  just  dealt  with,  and  to 
the  remainder  affix  the  next  number  in  the  dividend; 
then  4  into  25  will  go  5  times,  leaving  a  remainder  1. 
"Write  down  five  in  the  quotient;  next  affix  the  last  num- 
ber in  the  dividend  to  the  remainder;  then  4  into  12  w'ill 
go  twice,  leaving  2  as  a  remainder.  Write  down  the  2 
under  the  dividend.  Write  the  remainder  and  divisor 
separately. 

Examples. 

(1)  (2)  (3)  (4) 

245^3         4321^.5  6732^6  73215-^4 

3)245  5)4321  6)6732  4)73215 


67  7034  1117  166434 


o 


RULE  FOR  DIVISION  WHEN  THE  NUMBER  IS  GREATER 

THAN   EIGHT. 

Rule.  —  Place  the  divisor  and  dividend  thus  :  — 

divisor  )  dividend  (  quotient 

leaving  space  for  the  quotient  to  the  right  of  the  dividend.  Take 
off  from  the  left  hand  of  the  dividend  a  number  not  less  than 
the  divisor. 

Find  how  many  times  the  divisor  is  contained  in  this  number. 
Place  the  number  so  found  in  the  quotient ;  multiply  the  divisor 
by  the  quotient  and  bring  down  the  product  under  the  number 
taken  off  from  the  left  of  the  dividend,  and  subtract. 

On  the  right  of  the  remainder  bring  down  the  next  figure  in 
the  dividend.  Find  how  many  times  the  divisor  is  contained  in 
this  number.  If  it  is  less  than  the  divisor,  write  a  cipher  in  the 
quotient,  and  bring  down  another  figure  from  the  dividend  and 
affix  also  to  tlie  remainder ;  repeat  the  process  until  all  the 
figures  in  the  dividend  have  been  brought  down. 


32      Elementary  Arithmetic  of  the  Octimal  Notation. 

Example  1.  — Divide  1432  by  31. 

By  the  Rule.    31)1432(373       Method.  — 143  is  the  least 

113  number  taken  from  the  left 

OQ2  of  the  dividend  into  which 

cyrj  31  will  go;  for  if  31  be  X  3, 

the    product    will    be    113, 

which  is  less  than  the  num- 
ber taken.  Multiply  the  divisor  by  3;  place  the  product 
under  the  number  from  the  dividend  and  subtract;  the 
remainder  is  less  than  the  divisor;  .*.  place  the  multi- 
plier as  first  figure  in  the  quotient;  bring  down  the  next 
figure  from  the  dividend  and  affix  to  the  remainder. 
Find  how  many  times  the  divisor  is  contained  in  the 
new  dividend.  Place  the  multiplier  7  as  before  in  the 
quotient.  AVrite  down  the  product  under  the  new  divi- 
dend and  subtract.  There  is  a  remainder  23.  Write  it 
in  the  quotient,  and  the  divisor  under  it,  for  a  record. 


Examples. 

(1) 

14160^21 

23651  :  34 

>1) 14160(560 
125 

34)23651(552|i 
214 

146 
.  146 

225 
214 

...0 

111 
70 

« 

21 

Division.  ^^ 


(3) 
52765-^46 

32651-123 

46)52765(1103A 
46 

123)32651(245x^3 
246 

47 
46 

605 
514 

165 

162 

711 
637 

52 


34      Elementary  Arithmetic  of  the  Octimal  Notation. 


SECTION  3. 

DEFINITIONS. 

A  Unit  is  a  predeterminate  quantity  of  weight,  length, 
capacity,  or  value. 

For  purposes  of  calculation,  the  Unit  occupies  a  posi- 
tion immediately  to  the  left  of  a  point. 

Superior  Denominations  are  all  those  units  which  ap- 
pear to  the  left  of  the  point. 

Inferior  Denominations  are  all  those  units  which  ap- 
pear to  the  right  of  the  point. 

THE   POINT. 
A  Point  is  a  sign  so  placed  as  to  distinguish  and  to 
separate  Superior  Units  from  Inferior  Units  for  the  pur- 
pose of  determining   their  value.     Thus  1.1  means  one 
unit  plus  one  eighth  of  one  unit. 

THE   FRACTION. 

A  Fraction  is  an  undetermined  portion  of  a  Unit  re- 
maining after  units  have  been  subdivided  to  the  lowest 
convenient  denomination. 

Example.  —  Three  men  wish  to  apportion  4  dollars 
among  themselves  as  nearly  equal  as  possible.  The  cal- 
culation is  carried  out  by  the  rule  for  division;  thus, — 

$   (j;  Since  400  is  not  a  multiple  of  3,  and  the 

3)4.00         nearest  number  which  is  a  multiple  of  3  is  377, 

1  251  there  must  be  a  remainder  of  1  cent,  which 
must  be  dealt  with  in  a  manner  to  be  subse- 
quently shown. 


Reduction.  35 


REDUCTION. 

Reduction  is  carried  out  by  the  use  of  the  point,  termed 
Inspection;  thus  400  cents  =  $4.00. 

tons  lbs.  oz.  q.  pt.  in. 

32126  oz.  =  3.212.6       265  inches  Liquid  =  2.6.5 

yd.  ft.  in. 

265   "   Linear=2.6.5 
Thus  — 

E.   $  c 

17464.5.32     American  currency. 

miles  yd.  ft.  in. 

174.645.3.2     Linear  measure. 

yd.     ft.    in. 

1746.45.32     Square  measure. 

yd.    ft.      in. 

17.464.532     Cubic  measure. 

tank    pk.  q.  p.  i. 

17.464.5.3.2     Capacity  liquid  measure. 

ton  lb.  02.dr.sc.gr. 

1.746.4.5.3.2     Weights. 

days  hr.  m.    s. 

17.46.45.32     Time. 

Kevolution    °      '      " 

17.46.45.32     Circle. 


Table  of  Weights. 

10  Grains      make  1  Scruple. 
10  Scruples       "     1  Drachm. 
10  Drachms      "     1  Ounce. 
10  Ounces         "     1  Pound. 
1000  Pounds        "     1  Ton. 

Note. — The  pound  avoirdupois  is  the  unit  of  weight. 


36      Elementary  Arithmetic  of  the  Octimal  Notation. 
Table  of  Measures  of  Capacity. 

10  Minims         make  1  Cubic  Inch. 
10  Cubic  Inches    "      1  Pint. 
10  Pints  "      1  Quart. 

10  Quarts  "      1  Peck. 

1000  Pecks  "      1  Tank. 


Table  of  Linear  Measure. 

10  Inches  make  1  Foot. 

10  Feet  "      1  Yard. 

1000  Yards        "      1  Mile. 

Note.  —  Tlie  present  unit,  the  Inch,  is  to  be  retained. 


iii|iii|iii|in 


1  1  1     1  1  1 

1     1    1        1        1 

III     III 

1    1    1        1    1    1 

5 

6 

7 

10 

ONE- FOOT   EULE    ( SHOWN    IN   TWO    SECTIONS). 


Tables  of  Measures,  Money,  Etc.  37 

Currency. 

10  Mills  make  1  Cent. 
10  Cents      "     1  Bit. 
10  Bits         "      1  Dollar. 
10  Dollars  "     1  Eagle. 

Note.  — The  present  Dollar  is  the  unit  to  be  retained. 

Convenient  American  Coins. 

Gold.  Eagle=Eight  (10)  dollars To  be  minted. 

Silver.  Dollar  =  Etred  (100)  cents Present  dollar. 

••  Half-dollar  =  Four-et  (40)  cents  .       "    50  cents. 

"  Quarter-dollar  =  Two-et  (20)  cents       "    25  cents. 

"  Eighth-dollar  =  Eight  (10)  cents  .To  be  minted. 

Nickel.  Four  (4)  cents " 

Copper.  Two   (2)  cents " 

''  One    (1)  cent 


Value  of  Gold  Decimal  Coin  in  Octimal  Figures. 

Five         ($5)   pieces^- Five  (5)  dollars. 
Ten         ($10)  pieces  =  Et-two  (12)  dollars. 
Twenty  ($20)  pieces  =  Two-et  four  (24)  dollars. 


Measures  of  Time. 

100  Seconds  make  1  Minute. 
100  Minutes      '•'      1  Hour. 
100  Hours         "      1  Day. 


40       Elementary  Arithmetic  of  the  Octimal  Notation. 

Subdivisions  of  Circle. 

100"  (Seconds)  make  V  (Minute). 
100'  (Minutes)  "  1°  (Degree). 
100°  (Degrees)       "      1^  (Circle). 


COMPOUND   NUMBERS. 

Compound  Numbers  are  those  numbers  which  contain 
both  Superior  and  Inferior  units. 


COMPOUND   ADDITION. 

Rule.  —  Place  the  numbers  under  each  other,  units  under 
units,  eights  under  eights,  etc.,  so  that  the  points  shall  be  directly 
under  each  other.  Add  as  in  whole  numbers,  and  place  a  point 
in  the  result  immediately  under  the  points  in  the  numbers 
above. 


Compound  Multiplication.  41 

P:xample.  — Add  together  431.64,  56.200,  .0321. 

Bij  the  Rule.     431.64 

56.200 
.0321 


510.0721 


COMPOUND   SUBTRACTION. 

Rule.  —  Place  the  less  number  under  the  greater,  units  under 
units,  eights  under  eights,  etc.,  so  that  the  points  shall  be  directly 
under  each  other.  If  there  be  fewer  figures  in  the  upper  number 
to  the  right  of  the  point  than  there  are  figures  in  the  lower  num- 
ber to  the  right  of  the  point,  add  ciphers  to  the  upper  number 
until  the  numbers  of  figures  are  equal.  Then  subtract  as  in 
whole  number,  placing  a  point  immediately  under  the  point 
above. 

Example.  — Subtract  31.2652  from  53.21. 

By  the  Rule.     53.2100 
31.2652 


21.7226 


COMPOUND   MULTIPLICATION. 

Rule.  —  Multiply  the  numbers  together  as  if  they  were  whole 
numbers,  and  point  off  in  the  product  as  many  places  as  there 
are  in  both  the  multiplicand  and  multiplier.  If  there  are  not 
sufficient  figures,  prefix  ciphers. 

Example.  —  Find  the  product  of  .2X.3;  also,  1.4X2.3. 

By  the  Rule.     .2  1.4 

.3  2.3 

.06  44 

30 

3.44 


42       Elementary  Arithmetic  of  the  Octimal  Notation. 


COMPOUND   DIVISION. 

When  the  number  of  inferior  places  in  the  dividend 
exceeds  the  number  of  inferior  places  in  the  divisor. 

Rule.  —  Divide  as  in  whole  numbers,  and  point  off  in  the 
quotient  a  number  of  inferior  places  equal  to  the  excess  of  the 
number  of  inferior  places  in  the  dividend  over  the  number  of 
inferior  places  in  the  divisor ;  if  there  are  not  figures  sufficient, 
prefix  ciphers,  as  in  Multiplication. 

Example  1.  — Divide  23.006  by  2.35.    .0023006  by  2.35. 

Bij  the  Rule.     2.35)23.006(7.6         2.35). 0023006 (.00076 

2113  2113 


1 656  1656 

1  656  1656 


When  the  number  of  inferior  places  in  the  dividend  is 
less  than  the  number  of  inferior  places  in  the  divisor. 

Rule.  —  Affix  ciphers  to  the  dividend  until  the  number  of 
inferior  places  in  the  dividend  equals  the  number  of  inferior  places 
in  the  divisor  ;  the  quotient  up  to  this  point  of  the  division  will 
be  a  whole  number.  If  there  be  a  remainder,  and  the  division 
be  carried  on  further,  the  figures  after  this  point  in  the  quotient 
will  be  inferior  units. 

Example  2.  —  Divide  2300.6  by  .235. 

Bij  the  Rule.     .235)2300.600(7600. 

2113 

1656 
1656 

•    ••00 


Repetends.  '         43 

Rule.  —  Before  dividing,  affix  ciphers  to  the  dividend  to 
make  the  number  of  inferior  places  in  the  dividend  exceed  the 
number  of  inferior  places  in  tlie  divisor  by  three.  If  we  divide 
up  to  this  point,  the  quotient  will  contain  three  inferior  places. 

Example  3.  —  Divide  2301.2  by  .235  to  3  inferior  places. 

.235)2301.200000(7(301.502 
2113 


1662 

1656 

•••400 

235 

1430 

1421 

•700 

472 

206 


REPETENDS. 

The  sign  *  applied  to  a  number  means  that  the  num- 
ber is  recurring,  or  a  Repeater;  thus,  .2.  Repetends  are 
those  numbers  which  periodically  recur  in  Compound 
Division  and  may  be  extracted  to  the  lowest  required 
denomination  by  merely  affixing  them  to  the  quotient; 
thus,  4-^3  gives  the  quotient  1.2o;  and  the  quotient  may 
be  extended  thus,  1.252o2o. 


44      Elementary  Arithmetic  of  the  Octimal  Notation. 


SECTION  4. 

RATIO   AND   PROPORTION. 

The  relation  of  one  number  to  another  in  respect  of 
magnitude  is  called  Ratio. 

The  Ratio  of  one  number  to  another  is  denoted  by 
placing  a  colon  between  them.  Thus  the  ratios  of  3  to 
14  and  14  to  3  are  denoted  by  3:14  and  14:3. 

When  two  Ratios  are  equal,  they  are  said  to  form  a 
Proportion,  and  the  four  numbers  are  called  Proportionals. 
Thus  the  ratio  of  3  to  14  and  6  to  30  are  equal.  The 
Ratios  being  equal,  Proportionals  exist  among  the  num- 
bers 3,  6,  14,  30. 

The  existence  of  Proportion  between  the  numbers  3,  6, 
14,  30,  is  denoted  thus:  3:6::  14:30. 

If  four  numbers  be  proportionals  when  taken  in  a  cer- 
tain order,  they  will  also  be  proportionals  when  taken  in 
a  contrary  order;  for  instance,  3,  6,  14,  30,  are  propor- 
tionals, as  already  shown;  and  contrariwise,  30:14  ::  6:  3. 

The  two  numbers  which  form  a  Ratio  are  called  its 
terms;  the  former  term  is  called  the  Antecedent;  the  lat- 
ter, the  Consequent. 

Because  3X2=6,  and  14X2=30,  .'.  are  those  numbers 
proportional.  Again:  because  the  Antecedent  of  each 
when  multiplied  by  the  Consequent  of  the  other  produces 
a  like  number,  .*.  are  they  proportional. 

If  any  three  terms  of  a  proportion  be  given,  the  re- 
maining term  may  always  be  found.  For  since  in  any 
Proportion, 


Rule  of  Three.  45 

1st  term  X  4th  term  =  2d  term  X  3d  term, 
.-.  1st  term=2dX3d^4th 

2d  term^lstX4th^3d 

3d  term  =  lstX4th-^2d; 

4th  term=2dX3d^lst. 

Example   1.  —  Find  the  4th  term  in  the  proportion  3, 
6,  14. 

3:6;:  14.4th  term;  .'.  4th  term=6Xl4-^3  =  30,  Ans. 


RULE   OF  THREE. 

The  Rule  of  Three  is  a  method  l^y  which  we  are 
enabled,  from  three  numbers  which  are  given,  to  find  a 
fourth,  which  shall  bear  the  same  ratio  to  the  third  as 
the  second  to  the  first;  in  other  words,  it  is  a  rule  by 
which,  when  the  three  terms  of  a  proportion  are  given, 
we  can  determine  the  fourth. 

Rule.  —  Find  out,  of  the  three  quantities  which  are  given,  that 
which  is  of  the  same  kind  as  the  fourth  or  required  quantity,  or 
that  which  is  distinguished  from  the  other  terms  by  the  nature 
of  the  question.  Place  this  quantity  as  the  third  term  of  the 
proportion. 

Now  consider  whether,  from  the  nature  of  the  question,  the 
fourth  term  will  be  greater  or  less  than  the  third;  if  greater, 
then  put  the  larger  of  the  other  two  quantities  in  the  second 
term  and  the  smaller  in  the  first  term  ;  but  if  less,  put  the  larger 
in  tiie  first  term  and  the  smaller  in  the  second  term. 

Multiply  the  second  and  third  terms  together,  and  divide  by 
the  first.     The  quotient  will  be  the  answer  to  the  question. 


46      Elementary  Arithmetic  of  the  Octimal  Notation. 
Example  1.  —  If  oats  are  worth  $24  per  ton,  how  much 

ton  lb. 

can  be  bought  for  $50?     Since  1.000  is  of  the  same  kind 

ton  lb. 

as  the  required  term,  —  viz.,  tons,  —  we  make  1.000  the 
third  term.  Since  $50  will  buy  more  tons  than  $24,  we 
make  $50  the  second  term  and  $24  the  first  term: 

ton  lb. 

$24:$50::  1.000:4th  term. 

tons  lb. 

24-^50Xl.000=4th  term=2.000. 

Example  2.  —  A  has   a   contract  to  dig  a  ditch  in  2 

days.     He  has  4  men  employed,  but  at  present  progress 

it  will  take  4  days  to  complete  the  work.     How  many 

men  should  he  have  employed  to  finish  the  contract  on 

time? 

days  days  men 

2  :  4  ::  4  :  4th  term. 

2-^4X4  =  10,  the  number  of  men  required. 


PRACTICE. 

Simple  Practice. 

In  this  case  the  given  number  is  expressed  in  the  same 
denomination  as  the  unit  whose  value  is  given;  as,  for 
instance,  21  lb.  @  $1.20  per  lb. 

Rule.  —  Multiply  the  numbers  together  without  reference  to 
their  names,  and  point  off  the  terms  of  the  several  denominations 
in  the  product. 

Example.  —  43  lb.  of  sugar  @  7(J5  per  lb.  The  product 
will  be  cents;  .',  43X7=365  cents,  =  $3.65^ 


Interest.  47 

Compound  Practice. 

In  this  case  the  given  number  is  not  wholly  expressed 
in  the  same  denomination  as  the  unit  whose  value  is 
given;  as,  for  instance,  1  ton  235  lb.  @  $63  per  ton. 

Rule.  —  State  the  question  thus:  If  one  ton  cost  $G3.00,  what 
win  1,235  lb.  cost? 

Multiply  the  second  and  third  terms  together  without 
reference  to  their  names;  thus:  — 

1235X6300=10250;pj2l. 

From  the  right  of  the  product  cut  off  a  number  of 
figures  which  is  equal  to  the  number  of  inferior  units  in 
the  third  term.  The  remaining  figures  will  be  the  value 
required,  of  the  same  name  as  the  second  term.  Thus, 
$102.50({',  Ans. 


INTEREST. 

Interest  (Int.)  is  the  sum  of  money  paid  for  the  loan 
or  the  use  of  some  other  sum  of  money,  lent  for  a  certain 
time  at  a  fixed  rate;  generally  at  so  much  for  each  $100 
for  one  year. 

The  money  lent  is  called  the  Principal. 

The  interest  on  $100  for  a  year  is  called  The  Rate  per 
Cent. 

The  principal  +  the  interest  is  called  the  Amount. 

Interest  is  divided  into  Simple  and  Compound.  When 
interest  is  reckoned  only  on  the  principal  or  sum  lent, 
it  is  Simple  Interest. 


50      Elementary  Arithmetic  of  the  Octimal  Notation. 

When  the  interest  at  the  end  of  the  first  period,  instead 
of  being  paid  by  the  borrower,  is  retained  by  him  and 
added  as  principal  to  the  former  principal,  interest  being 
calculated  on  the  new  principal  for  the  next  period,  and 
this  interest,  again,  instead  of  being  paid,  is  retained  and 
added  on  to  the  last  principal  for  a  new  principal,  and  so 
on,  it  is  Compound  Interest. 

Simple  Interest. 

To  find  the  interest  on  a  given  sum  of  money  at  a 
given  rate  per  cent  for  a  year. 

Rule.  —  Multiply  the  princiiDal  by  the  rate  per  cent  and  divide 
the  product  by  100. 

Example.  —  Find  the  simple  interest  of  $240  for  one 
year,  at  7  per  cent  per  annum. 

By  the  Rule.     $240.00x7-100=$21.40. 

Compound   Interest. 

To  find  the  Compound  Interest  of  a  given  sum  of  money 
at  a  given  rate  per  cent  for  any  number  of  years. 

Rule.  —  At  the  end  of  each  year,  add  the  interest  for  that  year 
to  the  principal  at  the  beginning  of  it;  this  will  be  the  principal 
for  the  next  year ;  proceed  in  the  same  way  as  far  as  may  be 
required  by  the  question.  Add  together  the  interests  so  arising 
in  the  several  years,  and  the  result  will  be  the  compound  interest 
for  the  given  period. 

Example.  —  Find  the  Compound  Interest  and  Amount 
of  $600  for  3  years,  at  6  per  cent  per  annum. 


Practice.  51 

600X6-100=^44  Int.  1st  yr. 
600  +  44  =  644x6-h100=$47..30  Int.  2d  yr. 
644  +  47.30=713.30X6^100=$53.0420  Int.  2d  yr. 
713.30  + 53.0420= Ans.  $766.34.20. 
.-.  Compound  interest  =  $53.0420  +  $47.30-f  .1?44=$166.- 
3420. 

Amount,  $600  + $166.3420=1766.342. 


52      Elementary  Arithmetic  of  the  Octimal  Notation. 


SECTION  5. 

SQUARE   ROOT. 

The  Square  of  a  given  number  is  the  product  of  that 
number  multiplied  by  itself.  Thus  6X6,  or  44,  is  the 
square  of  6,  or  44=6^ 

The  Square  Root  of  a  given  number  is  a  number  which, 
when  multiplied  by  itself,  will  produce  the  given  num- 
ber. 

The  Square  Root  of  a  number  is  denoted  by  placing 
the  sign  ;/  before  the  number.  Thus  -|/44  denotes  the 
square  root  of  44,  so  that  y^44=6.  The  sign  ^  placed 
above  the  number  a  little  to  the  right  denotes  that  the 
number  is  to  be  squared;  thus,  6^=44. 

Note.  —  The  squares  of 
all  those  numbers  which 
are  in  geometrical  progres- 
sion appear  with  the  order 
of  their  indices  reversed, 
and  their  roots  may  be 
extracted  by  mere  inspec- 
tion. 

A  square  figure  Avill  al- 
ways contain  the  square 
of  a  number  of  square 
units. 

The  square  root  of  a  number  is  one  side  of  a  square  with  area 
equal  to  the  number. 


Table  1. 

Table  2. 

1  =  1/1 

1  = 

1/1 

2=  4 

• 

2- 

4 

3=  11 

• 

4  — 

26 

4=  20  . 

16— 

106 

5-  31 

26  = 

406 

6-  44 

•  • 

40 

2606 

7-  61 

166— 

10606 

10=  100 

266— 

40606 

Square  Root.  53 

.*.  A  square  containing  44  square  units  has  a  side  containing 
6  linear  units  of  the  same  vahie  as  one  side  of  each  of  the  44 
S(juare  units  contained  in  the  given  square. 


RULE   TO   EXTRACT  THE   SQUARE  ROOT  OF  A   GIVEN 

NUMBER. 

Rule.  —  Place  a  dot  over  the  units'  place  of  the  given  number  ; 
and  thence  over  every  second  figure  to  the  left  of  that  place ; 
and  thence  over  every  second  figure  to  the  right,  when  the  num- 
ber contains  inferior  units,  annexing  a  cipher  when  the  number 
of  inferior  places  is  odd ;  thus  dividing  the  number  into  periods. 
The  number  of  dots  over  the  superior  and  inferior  units  respec- 
tively will  show  the  superior  and  inferior  places  in  the  root. 

From  Table  1  take  the  index  which  is  nearest  (but  less)  to 
the  first  period,  and  write  its  root  in  the  first  place  in  the  root. 
Now  consider  whether  the  first  period  is  nearer  the  index  already 
taken,  or  nearer  the  index  which  comes  next  in  the  table.  If 
nearer  the  index  taken,  try  a  number  less  than  four  for  the 
second  place  in  the  root ;  now  square  the  number  for  trial.  If 
the  product  is  greater  than  the  given  number,  try  a  number 
which  is  less ;  repeat  the  operation  until  all  the  places  in  the 
root  are  filled. 

FlxAMPLE  1.  —  Find  the  Square  Root  of  1261.04. 

32.2  =  i/l26i.04 

Sy/ll  4|/  20       Method.  — From  Table  1  take  the 

Index  11;  the  next  one,  20;  apply 
the  root  of  the  11  to  first  place;  first 
period  is  near  11;  .*.  try  32^,  also 
33';  the  product  1244  is  nearest  the 
given    number;    .*.    write    2   in    the 

1244  1331      root;    32^  is  much  nearer  the  given 

number  than  33";  .•.  try  322. 


32 

33 

32 

33 

64 

121 

116 

121 

54      Elementary  Arithmetic  of  the  Octimal  Notation. 


322 

322 

644 
644 
1166 


322X322  =  126104. 

Square  root  of  1261.04=32.2. 

A  RECTANGLE. 


A  mixed  number  of  square  units  will  always 
126104        be  contained  in  a  Parallelogram,  one  of  whose 
dimensions  shall  be  a  root  number,  as  in  Table  2. 

Rule.  —  To  conform  the  given  number  to  a  parallelogram, 
divide  the  given  number  by  the  selected  dimension.  The  quotient 
will  be  a  perpendicular.  The  divisor  and  the  quotient  will  be  the 
required  dimensions. 

Example  1.  —  Given  one  side  and  the  area  of  a  Paral- 
lelogram, to  find  the  other  side,  one  side  to  be  20,  the 
area  to  be  541. 

By  the  Rule.     541^20=26.04; 
.*.  one  dimension =20; 


u 


u 


=  26.04. 


o 

o 


<M 


Area,  541  Units. 


26.04 


THIS    DIAGRAM    IS    ONE    EIGHTH    ACTUAL   DIMENSIONS. 


oo 

b 


Cube  Boot.  55 

Example  2. — Conform  541  to  the  required  Parallelo- 
gram, one  side  to  be  40. 

Bij  the  Rule.     541-^40=13.02; 
.'.  one  dimension=40; 

=  13.02. 


Area,  541  Units. 


40.00 


THIS    1)[A(4RA>[    IS    ONE    EIGHTH    ACTUAL    DIMENSIONS. 


CUBE  ROOT. 

The  Cube  of  a  given  number  is  the  product  which  arises 
from  multiplying  that  number  by  itself,  and  then  mul- 
tiplying the  result  again  by  the  same  number.  Thus 
6X6X6,  or  330,  is  the  cube  of  6;  or  330=61 

The  Cube  Root  of  a  given  number  is  a  number  which, 
when  multiplied  into  itself,  and  the  result  is  again  mul- 
tiplied by  it,  will  produce  the  given  number.  Thus  6  is 
the  Cube  Root  of  330;  for  6X6=44,  and  44X6=330. 

The  Cube  Root  of  a  number  is  denoted  by  placing  the 
sign  f  before  the  number.  Thus  ]^330  denotes  the 
Cube  Root  of  330;  so  that  ]f  330=6. 


Table  1. 

Table  2. 

1-fa 

1-1 

3/1 

2-   10 

2- 

16 

3-  33 

4- 

100 

4-  100 

10- 

iooo 

5-  175 

20- 

16000 

6=  330 

40= 

106006 

7-  527 

100- 

1006006 

10-  1000 

200- 

16006006 

56      Elementary  Arithmetic  of  the  Octimal  Notation. 


Note. — The  Cubes 
of  all  those  numbers 
which  are  in  geomet- 
rical progression  oc- 
cur in  ciphers  with 
the  index  1.  Their 
roots  may  be  extracted 
by  mere  inspection. 

All  the  indices  will 
reappear   when    their 
roots  are  bisected  continuously. 

Cube  Root  is  the  number  of  units  of  measure  contained  in  a 
line  which  is  one  edge  of  a  Cube. 

The  Cube  Root  of  a  number  can  only  be  found  when  the  given 
number  is  the  cube  of  the  number  of  units  in  the  base. 

RULE  FOR  EXTRACTING  THE  CUBE  ROOT  OF  A  GIVEN 

NUMBER. 

Rule.  — Place  a  dot  over  the  units'  place  of  the  given  number, 
and  thence  over  every  third  figure  to  the  left,  and  thence  over 
every  third  figure  to  the  right,  when  the  number  contains  inferior 
units,  aflixing  one  or  two  ciphers  when  necessary  to  make  the 
number  of  inferior  places  a  multiple  of  three,  thus  dividing  the 
given  numbers  into  periods.  The  number  of  dots  over  the  Avhole 
number  and  inferior  units,  respectively,  will  show  the  whole 
number  and  inferior  places  in  the  root. 

Place  the  sign  |^'  over  the  given  number.  Point  over  the  same 
number  of  places  to  the  left  of  the  sign  the  same  number  of  dots 
that  there  are  periods  in  the  given  number,  in  the  form  of  a  divisor. 

Find  among  the  indices  the  number  which  is  nearest  (but 
below)  the  number  in  the  first  period,  and  place  the  correspond- 
ing root  number  in  the  first  place  in  the  divisor. 

Now  consider  whether  the  index  just  applied  is  nearer  the 
given  number  than  is  the  index  which  is  nearest  above  or  greater 
than  the  given  number. 


Cube  Root. 


57 


If  the  index  applied  is  the  nearest,  add  a  number  which  is  less 
than  four,  now  cube  the  new  number,  and  if  the  product  be  less 
than  the  given  number,  writedown  the  figure  in  the  second  place 
in  the  root,  but  if  it  be  greater,  trj'  a  lower  number.  This  opera- 
tion is  to  be  repeated  until  all  the  places  in  the  root  are  filled. 


THIS    CUT   SHOWS    ONE   EIGHTH    OF    THE    ACTUAL    DIMENSIONS. 


60      Elementary  Arithmetic  of  the  Octimal  Notation, 
Example.  —  Extract  the  Cube  Root  of  16606.305, 

23.5|/16606.305 


By  the  Rule.    2|/10         Zy'66 


23 

24 

235 

23 

24 

-235 

71 

120 

1421 

46 

50 

727 

551 

620 
24 

472 

23 

60111 

2073 

3100 

235 

1322 

1440 

360555 

15313 

17500 

220333 
140222 

Method. — From  Table 
1  take  the  index  10;  ap- 
ply its  root  to  the  first 
place;  the  index  applied 
is  nearer  the  first  period 
than  33;  .'.  try  23^,  also 
24^,  then  write  3  in  the 
second  place.  The  given 
number  is  nearer  24^;  .*. 
try  2351 


16606305 


RECTANGLE  PARALLELOPIPED. 

A  mixed  number  of  cubical  units  will  always  be  con- 
tained in  a  Parallelopiped,  two  of  whose  dimensions  shall 
be  a  root  number  in  Table  2. 

Rule.  —  To  conform  a  given  number  to  a  Parallelopiped,  mul- 
tiply the  selected  dimensions  together  and  divide  the  given  num- 
ber by  the  product.  The  quotient  will  be  the  third  dimension  of 
a  Parallelopiped. 


Example  1.  —  Conform  6340  to  a  Parallelopiped. 


Eectan(jle  Parallelopiped, 

By  the  Rule.     Selected  dimensions  =  20  each; 
•.  20X20  =  400  =  area  of  one  side; 
6340-^400= third  dimension  =  14.7; 
.-.  one  dimension  =  20; 
Second  dimension  =  20; 
Third  dimension  =  14.7. 


61 


THIS   CUT    SHOWS    ONE   EIGHTH    OF   THE    ACTUAL   DIMENSIONS. 


Example  2.  —  Selected  dimensions  are:  one  dimension, 
10;  one  dimension,  40; 

.-.  10X40=400=area  of  one  side; 
6340 -^400= third  dimension  =  14.7; 
.'.  one  dimension  =  10; 
Second  dimension =40; 
Third  dimension=14.7. 


62      Elementary  Arithmetic  of  the  Octimal  Notation. 

DERIVATION  OF  RADIX. 

The  Radix  is  suggested  by  the  number  of  parts  into 
which  the  Cube  may  be  subdivided,  Tvhen  all  its  dimen- 
sions are  bisected.  Hence  the  number  of  symbols  in  the 
Octimal  Notation  permit  the  same  mechanical  treatment 
as  does  the  Cube  or  any  other  regular  solid;  .'.  since  the 
third  mechanical  subdivision  of  the  Cube  results  in  10 
Cubes,  each  a  counterpart  of  the  original  in  the  ratio  of 
10  to  1,  therefore  will  the  third  bisection  of  the  radix  re- 
sult in  symbols  the  counterpart  of  the  original  in  the 
ratio  of  10  to  1. 


ELEVATION  OF  DENOMINATIONS. 

Whole  numbers  are  raised  to  the  next  higher  denomina- 
tion by  affixing  a  cipher;  thus,  35X10=350. 

Whole  numbers  are  reduced  to  the  next  lower  denomina- 
tion by  introducing  a  point;  thus,  35-^10  =  3.5. 

.•.  let  A  represent  a  cube,  and  a,  a  cube,  then  if  the 


weight  of  A^ 


:35  lb.,  a^ 


:3.5  lb 


Calculation  by  Construction.  63 

ELEVATION  OF   CUBES   AND   THEIR  ROOTS. 

For  each  time  a  Root  is  multiplied  by  two,  raise  cube 
number  one  denomination  by  affixing  a  cipher. 

For  each  time  a  Root  is  bisected,  reduce  its  cube  num- 
ber one  denomination  by  pointing  off  one  figure  from  the 
right  of  the  whole  number. 

CALCULATION  BY  CONSTRUCTION. 

A  Cube,  Sphere,  or  Parallelopiped  may  be  reduced  to 
convenient  limits  for  calculation  by  simply  bisecting  its 
diameter  to  the  lowest  convenient  denomination. 

Rule. — For  each  time  the  three  dimensions  are  bisected, 
write  down  one  cipher,  and  for  the  last  place  apply  the  index  1, 
or  .r.  When  the  value  of  the  index  is  found,  replace  the  index 
by  that  value. 

Example.  —  A  cube  of  bricks  is  140.0  feet  long  at  base. 
How  many  bricks  does  it  contain?  and  what  is  its  value 
at  $7.00  per  etand  bricks? 


^-^ 


By  the  Rule. 
2)140.0 

2)60.0  i  Number,  12,530,000. 

Ans.  ' 


2)30.0  (  Value,  12,530,000X$7.00=$112,550.00. 

2)14.0 

6.0=6.0  ft.  =  Index. 

The  Index  is  found  to  contain  1253  bricks;  therefore  to 
the  sum  so  found  affix  four  ciphers. 


64      Elementary  Arithmetic  of  the  Octimal  Notation. 


SECTION  6. 

LONGITUDE  AND   TIME. 

The  Earth's  Circle  is  subdivided,  commencing  opposite 
the  meridian  of  Greenwich. 

The  Dial  is  subdivided  to  correspond  from  midnight. 
Longitude  is  therefore  always  west. 
Difference  in  time  gives  Longitude. 

Example.  —  Meridian  of  Departure =0.  The  sun  passes 
overhead  40  ^^'-  2V  later.  What  is  your  Longitude? 
Ans.  40°  (degrees)  21'  (minutes). 

TO  DETERMINE  AN  ANGLE. 


>\ 


Example.  —  Required  the  Angle  A  0  B. 
Ans.  Angle  =  12  degrees. 


Area  of  Circle  in  Terms  of  Square  Units.  65 

Rule.  —  At  point  of  intersection  raise  a  perpendicular.  Bisect 
the  right  angle.  The  value  of  the  new  angle -^10.  Should  the 
angle  exceed  10,  bisect  again  and  continue  to  the  ratio  of  1  to  10, 
repeating  the  operation  if  necessary. 

When  the  Angle  is  less  than  10,  the  operation  is  similar. 


AREA  OF   CIRCLE  IN  TERMS   OF   SQUARE   UNITS. 

The  diameter  of  a  circle  is  to  its  circumference  as  1 
is  to  3.1. 

The  square  of  the  diameter  of  a  circle  is  to  its  area  as 
1  is  to  .62. 

.'.  Multiply  the  square  of  the  diameter  by  .6222. 

Example.  —  Required  the  area  of  a  circle  whose  di- 
amter  is  16.2  inches. 

By  the  Rule.     16.2^  =  313.04 X.6222  =  237.421710  inches. 


66      Elementary  Arithmetic  of  the  Octimal  Notation. 


SECTION  7. 

MISCELLANEOUS. 

1  Meter  =  47. 2753412  Inches,  nearly. 

.•.  A  Centimeter  is  to  an  Inch  as  1:2.43;  /.  InchesX 
2.43  =  number  of  Centimeters. 

An  Inch  is  to  a  Centimeter  as  1:.311;  .'.  Cm.X.311  = 
number  of  Inches. 

Note.  —  This  comparison  is  sufficiently  accurate  for  all  prac- 
tical pm'poses  in  measurements  up  to  one  foot. 

Example.  —  Required  the  value  of  6.2  Inches  in  Centi- 
meters. 


m. 


By  the  Rule.     6.2X2.43  =  17.723  Centimeters. 


TRANSFORMATION  OF   NUMBERS. 

Decimal  numbers  may  be  transformed  into  Octimal 
numbers. 

Rule. —  Divide  by  eight  continuously  ;  the  remainders  will  be 
the  Octimal  figures. 

Example.  —  1631  Decimal  =  3137  Octimal. 

By  the  Rule.     8)1631 

8)203(7  .  .   , 
Note.  —  The  operation  is  carried 

8)25(3      on  in  the  decimal  system. 


Decimal  Fractions.  67 

Octimal   numbers   may  be   transformed   into  Decimal 
numbers. 

Rule.  —  Divide  by  et-two  continuously  ;  the  remainders  will  be 
the  Decimal  numbers. 

Example.  — 15356  Octimal=6894  Decimal. 
By  the  Rule.     12)15356 


12)1261(4 
12)104(11=9 
6)10=8 


DECIMAL    FRACTIONS.  — TO   TRANSFORM   INTO    IN- 
FERIOR UNITS. 

Rule.  —  Multiply  by  eight  continuously.  The  figures  to  the 
left  of  the  point  in  each  product  will  be  the  Octimal  Inferior 
Units  required. 

Example.  —  .125  =  .l. 

By  the  Rule.     .125 

8 


1.000 


Note.  —  When  all  the  figures  in  the   i)roduct   are   below   the 
point,  supply  a  cipher  in  the  units'  place. 

Example.  —  .0625  =  .04. 

By  the  Rule.    .0625 

8 


0.5000 
8 

4.0000 


70      Elementary  Arithmetic  of  the  Octimal  Notation. 

DECIMAL  NUMBERS  AND  OCTIMAL  INFERIOR 

UNITS. 

Addition. 

Rule.  —  Add  the  octimal  inferior  units  together,  carrying 
eights.  Carry  the  superior  units,  and  add  to  the  decimal  num- 
bers above  the  point.     Continue  in  the  decimal  system. 

Dec.  Oct.  Dec.  Vulgar  Fractions. 

Example. —  29.3  inches=29  +  |  inches. 


29.6 

u 

-29  +  1 

a 

29.23 

u 

-29  +  i 

a 

4-  3 
+  "6  4' 

29.76 

u 

-29  +  1 

a 

4-    ^ 

+  61^ 

118.31  118  +  1  inches  + ^x. 


Multiplication. 

Dec  Oct.  Dec.  Vul.  Frac. 

118.31x4  =  118+1  +  ^^X4 
4  4 

473.44  473  +  i  +  e\ 


List  of  Educational  Publications 

...OF... 

THE  WHITAKER  &  RAY  COMPANY 

San  Francisco 
Complete  Descriptive  Circular  sent  on  application 


Net  Prices 

Algebraic  Solution  of  Equations— Andre  and  Buchanan     -        -        -  $0  80 

An  Aid  to  the  Lady  of  the  Lake,  etc.— J.  \V.  Graham  -        -        .  25 

Amusing  Geography  and  Map  Drawing— Mrs.  L.  C.  Schutze  -        -  1  00 

Brief  History  ot  California— Hittell  and  Faulkner    -        -        -        .  50 

Current  History— Harr  Wagner 25 

Civil  Government  Simplified— J.J.  Duvall 25 

Complete  Algebra— J.  B.  Clarke 1  00 

Elementary  Exercises  in  Botany— V.  Rattan 75 

Grammar  by  the  Inductive  Method— W.  C.  Doub      -       -        -        -  25 

Heart  Culture — Emma  E.  Page       -       -       - 75 

How  to  Celebrate— J.  A.  Shedd 25 

Key  to  California  State  Arithmetic— A.  M.  Armstrong         -        -  1  00 

Key  to  West  Coast  Botany— V.  Rattan 1  00 

Lessons  Humane  Education — Emma  E.  Page — per  part  ...  25 

Lessons  in  Nature  Study— Jenkins  and  Kellogg    -       -       -        -        -  1  00 

Lessons  in  Language  Work — Susan  Isabel  Frazee  ...       -  50 

Manual  of  School  Law— -J.  W.  Anderson 1  25 

Matka— David  Starr  Jordan 75 

Moral  Culture  as  a  Science— Bertha  S.  Wilkins           .       -       -       -  1  00 

Nature  Stories  of  the  North-West— Herbert  Bashford   -       -       -  50 

New  Essentials  of  Book-keeping— C.  W.  Childs           ....  75 

Orthoepy  and  Spelling— John  W.  Imes — per  part        ....  20 

Poems  for  Memorizing — Alice  R.  Power          ......  60 

Paper  and  Cardboard  Construction— A.  H.  Chamberlain       -       -  75 

Pacific  History  Stories— Harr  Wagner 50 

Pacific  Nature  Stories — Harr  Wagner 50 

Patriotic  Quotations — Harr  Wagner 40 

Readings  from  California  Poets — Edmund  Russell         ...  25 

Science  Record  Book— Josiah  Keep  ----.-..  50 

Shells  and  Sea  Life— Josiah  Keep         --.....  50 

Stories  of  Oregon — Eva  E.  Dye 50 

Supplement  to  State  History — Harr  Wagner 25 

Spanish  in  Spanish      -_-- 125 

Spanish  Phonography — I.  I.  Ferry 1  00 

Story  of  Evangeline — L.  H.\incent 25 

Stories  of  Our  Mother  Earth— H.W.  Fairbanks      ...        -  50 

Studies  in  Entomology — H.M.Bland 50 

Study  of  the  Kindergarten  Problems— F.  L.  Burk         ...  50 

Tales  of  Discovery  on  Pacific  Slope — M.  G.  Hood    -        -        .        -  50 

Tales  of  Philippines— R.  Van  Bergen 50 

Topical  Analysis  of  United  States  History— C.  W.  Childs     -        .  75 

Topical  Discussions  of  American  History — W.  C.  Doub     -        -  60 

Toyon  Holiday  Recitations — AUie  M.  Felker 35 

West  Coast  Shells— Josiah  Keep 175 

LATEST    ISSUES 

Browne's  Graded  Mental  Arithmetic  —  Frank  J.  Browne          -        -  30 

New  Pacific  Geography  —  California  Edition  —  Harr  Wagner    -  1  00 
Practical  Aids  to  Literature,  No.  i— Irving's  Selections— J.  W.  Graham  25 


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